Each choice of 2 distinct vertices yields two triangles (one inside the nonagon, and one outside). Since , and , equilateral triangles inside the nonagon among every third vertex use three vertices of the nonagon. So, there are ways to choose two vertices. Among them, there are three distinct sets of three vertices, each three apart. There are ways of choosing each such set of three vertices. Hence, there are distinct equilateral triangles formed where two vertices are vertices of the regular nonagon.